[{"id":247260,"date":"2026-06-27T13:12:03","date_gmt":"2026-06-27T18:12:03","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247260"},"modified":"2026-06-27T13:13:36","modified_gmt":"2026-06-27T18:13:36","slug":"decimal-period","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/decimal-period\/","title":{"rendered":"When will the decimals in a\/b repeat?"},"content":{"rendered":"

The previous post looked at how many digits are in the reduced fraction for the\u00a0n<\/em>th harmonic number. I was curious about how long the cycle of digits in a harmonic number might be.<\/p>\n

I wrote about the period length for the digits of fractions almost a decade ago<\/a>. This post includes code so I can apply it to harmonic demoninators.<\/p>\n

\r\nfrom sympy import lcm, factorint, n_order\r\n\r\ndef period(n):\r\n    factors = factorint(n)\r\n    exp2 = factors.get(2, 0)\r\n    exp5 = factors.get(5, 0)\r\n    r = max(exp2, exp5)\r\n\r\n    d = n \/\/ (2**exp2 * 5**exp5)\r\n    s = 1 if d == 1 else n_order(10, d)\r\n    return (r, s)\r\n<\/pre>\n
This function returns two numbers: r<\/em> is the number of non-repeating digits at the beginning and s<\/em> is the length of the repeating part. <\/p>\n
The following code<\/p>\n
\r\nfrom functools import reduce\r\n\r\ndef lcm_range(n):\r\n    return reduce(lcm, range(1, n + 1))\r\n\r\nprint( period( lcm_range(50) ) )\r\n<\/pre>\n
prints (5, 1275120) meaning that 1\/lcm(1, 2, 3, …, 49, 50) has five non-repeating digits following by 1,275,120 digits that repeat ad infinitum<\/em>. And so the decimals in the expansion of H<\/em>50<\/sub> go have a cycle length of 1,275,120.<\/p>\n","protected":false},"excerpt":{"rendered":"
The previous post looked at how many digits are in the reduced fraction for the\u00a0nth harmonic number. I was curious about how long the cycle of digits in a harmonic number might be. I wrote about the period length for the digits of fractions almost a decade ago. This post includes code so I can […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[94],"class_list":["post-247260","post","type-post","status-publish","format-standard","hentry","category-math","tag-number-theory"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"When will the decimals in a\/b repeat?","description":"How to find out the length of the cycle of repeating digits in a fraction a\/b when b is so large that just computing a\/b and looking isn't practical.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/decimal-period\/","robots":"max-image-preview:large","keywords":"number theory","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"When will the decimals in a\/b repeat?","og:description":"How to find out the length of the cycle of repeating digits in a fraction a\/b when b is so large that just computing a\/b and looking isn't practical.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/decimal-period\/","article:published_time":"2026-06-27T18:12:03+00:00","article:modified_time":"2026-06-27T18:13:36+00:00","twitter:card":"summary","twitter:title":"When will the decimals in a\/b repeat?","twitter:description":"How to find out the length of the cycle of repeating digits in a fraction a\/b when b is so large that just computing a\/b and looking isn't practical.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247260","title":null,"description":"How to find out the length of the cycle of repeating digits in a fraction a\/b when b is so large that just computing a\/b and looking isn't practical.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-06-27 17:57:22","updated":"2026-06-27 18:13:36","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null},"aioseo_breadcrumb":"
\n\t\t\tHome<\/a>\n\t\t<\/span>»<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>»<\/span>\n\t\t\tWhen will the decimals in a\/b repeat?\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"When will the decimals in a\/b repeat?","link":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/decimal-period\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247260","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=247260"}],"version-history":[{"count":2,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247260\/revisions"}],"predecessor-version":[{"id":247262,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247260\/revisions\/247262"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=247260"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=247260"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=247260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":247253,"date":"2026-06-27T07:51:42","date_gmt":"2026-06-27T12:51:42","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247253"},"modified":"2026-06-27T07:51:42","modified_gmt":"2026-06-27T12:51:42","slug":"height-of-harmonic-numbers","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/height-of-harmonic-numbers\/","title":{"rendered":"Height of harmonic numbers"},"content":{"rendered":"
The previous post<\/a> looked at writing the harmonic numbers as reduced fractions and estimating the number of digits in the numerator and denominator based on asymptotics. This is a follow up post with plots.<\/p>\n
We’ll choose our base\u00a0b<\/em> to be 2. And we’ll look at the total number of bits in both the numerator and denominator, which we will use as the height<\/a> of the fractions.<\/p>\n
First, let’s look at the actual and estimated heights, using the estimates from the previous post.<\/p>\n
<\/p>\n
Next let’s look at the difference between the actual and estimated heights.<\/p>\n
<\/p>\n
In the previous post I looked at n<\/em> = 50, which was kind of a lucky choice, the error being smaller than usual. I had also looked at, but didn’t publish, n<\/em> = 100, which would be an unlucky choice.<\/p>\n
Finally, let’s look at the relative<\/em> error in the estimates, and plot over a larger range of n<\/em>.<\/p>\n
<\/p>\n
The error goes to zero, as predicted by the asymptotic estimates. And it goes noisily, which you’d expect since the heights are related to the distribution of primes.<\/p>\n","protected":false},"excerpt":{"rendered":"
The previous post looked at writing the harmonic numbers as reduced fractions and estimating the number of digits in the numerator and denominator based on asymptotics. This is a follow up post with plots. We’ll choose our base\u00a0b to be 2. And we’ll look at the total number of bits in both the numerator and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[94],"class_list":["post-247253","post","type-post","status-publish","format-standard","hentry","category-math","tag-number-theory"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Height of harmonic numbers","description":"Plotting the actual and estimated number of bits in the nth harmonic number.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/height-of-harmonic-numbers\/","robots":"max-image-preview:large","keywords":"number theory","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Height of harmonic numbers","og:description":"Plotting the actual and estimated number of bits in the nth harmonic number.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/height-of-harmonic-numbers\/","article:published_time":"2026-06-27T12:51:42+00:00","article:modified_time":"2026-06-27T12:51:42+00:00","twitter:card":"summary","twitter:title":"Height of harmonic numbers","twitter:description":"Plotting the actual and estimated number of bits in the nth harmonic number.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247253","title":null,"description":"Plotting the actual and estimated number of bits in the nth harmonic number.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-06-27 12:35:46","updated":"2026-06-27 17:57:48","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null},"aioseo_breadcrumb":"
\n\t\t\tHome<\/a>\n\t\t<\/span>»<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>»<\/span>\n\t\t\tHeight of harmonic numbers\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"Height of harmonic numbers","link":"https:\/\/www.johndcook.com\/blog\/2026\/06\/27\/height-of-harmonic-numbers\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247253","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=247253"}],"version-history":[{"count":4,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247253\/revisions"}],"predecessor-version":[{"id":247257,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247253\/revisions\/247257"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=247253"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=247253"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=247253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":247251,"date":"2026-06-26T20:51:03","date_gmt":"2026-06-27T01:51:03","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247251"},"modified":"2026-06-27T07:59:00","modified_gmt":"2026-06-27T12:59:00","slug":"writing-down-harmonic-numbers","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/06\/26\/writing-down-harmonic-numbers\/","title":{"rendered":"Writing down harmonic numbers"},"content":{"rendered":"
The\u00a0n<\/em>th harmonic number is the sum of the reciprocals of the first\u00a0n<\/em> positive integers.<\/p>\n
H<\/em>n<\/em><\/sub> = 1 + 1\/2 + 1\/3 + 1\/4 + \u2026 + 1\/n<\/em><\/p>\n
The product of all the denominators is\u00a0n<\/em>!, so you could write H<\/em>n<\/em><\/sub> as a fraction<\/p>\n
H<\/em>n<\/em><\/sub> = p<\/em>\/q<\/em><\/p>\n
where\u00a0p<\/em> =\u00a0n<\/em>! H<\/em>n<\/em><\/sub> is an integer and q<\/em> =\u00a0n<\/em>!.<\/p>\n
While p<\/em>\/q<\/em>\u00a0is a<\/em> way to write H<\/em>n<\/em><\/sub> as a fraction, it’s not the most efficient because\u00a0p<\/em> and\u00a0n<\/em>! will have common factors.<\/p>\n
If we write H<\/em>n<\/em><\/sub> as a reduced fraction, the denominator will be the least common multiple of the integers 1 through\u00a0n<\/em>. That number is asymptotically exp(n<\/em>). That estimate follows from the prime number theorem.<\/p>\n
So for large n<\/em> the denominator will be roughly exp(n<\/em>), and in base\u00a0b<\/em> it would have around<\/p>\n
n<\/em>\/log(b<\/em>)<\/span><\/p>\n
digits.<\/p>\n
The numerator will be exp(n<\/em>) H<\/em>n<\/em><\/sub>, and since H<\/em>n<\/em><\/sub> is asymptotically log(n<\/em>) + \u03b3, the numerator for large\u00a0n<\/em> will be roughly<\/p>\n
exp(n<\/em>) (log(n<\/em>) + \u03b3)<\/p>\n
and will have around<\/p>\n
(n<\/em> + log log(n<\/em>) ) \/ log(b<\/em>)<\/span><\/p>\n
digits.<\/p>\n
Let’s see how well our asymptotic estimates work for\u00a0n<\/em> = 50. The 50th harmonic number is<\/p>\n
H<\/em>50<\/sub> = 13943237577224054960759 \/ 3099044504245996706400.<\/p>\n
This fraction has 23 digits in the numerator and 22 in the denominator. We would have predicted around<\/p>\n
(50 + log(log(50)))\/log(10) = 22.3<\/p>\n
digits in the numerator and<\/p>\n
50\/log(10) = 21.7<\/p>\n
digits in the denominator.<\/p>\n
Let’s try a larger example, looking at the 1000th harmonic number in binary. We’ll use the following Python code.<\/p>\n
from fractions import Fraction\r\n\r\ndef bits(n):\r\n    H = sum(Fraction(i, i+1) for i in range(1, n+1))\r\n    p, q = H.numerator, H.denominator\r\n    # subtract 2 because bin returns a string starting with 0b.\r\n    return len(bin(p)) - 2, len(bin(q)) - 2\r\n\r\nprint(bits(1000))\r\n<\/pre>\n
This returns 1448 and 1438. We would have estimated<\/p>\n
(1000 + log(log(1000)))\/log(2) = 1445.4<\/p>\n
bits in the numerator and<\/p>\n
1000\/log(2) = 1442.7<\/p>\n
bits in the denominator.<\/p>\n
Update<\/strong>: See the next post<\/a> for plots as a function of\u00a0n<\/em>.<\/p>\n
Related posts<\/h2>\n